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A094687 Convolution of Fibonacci and Jacobsthal numbers. 3
0, 0, 1, 2, 6, 13, 30, 64, 137, 286, 594, 1221, 2498, 5084, 10313, 20858, 42094, 84797, 170582, 342760, 688105, 1380390, 2767546, 5546037, 11109786, 22248228, 44542825, 89160674, 178442742, 357081901, 714481614, 1429477456, 2859786953 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Also convolution of A008346(n-1) and A000079(n).

Also difference of Fibonacci and Jacobsthal numbers shifted left: a(n) = A000045(n+1) - A001045(n+1). - David Callan, Jul 22 2008

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,2,-3,-2).

FORMULA

G.f.: x^2/((1-x-x^2)*(1-x-2*x^2)).

a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) - 2*a(n-4).

a(n) = Sum_{k=0..n} A000045(k)*A001045(n-k).

a(n+1) = a(n) + 2*a(n-1) + A000045(n). - Philippe Deléham, Mar 06 2013

a(n) = J(n+1) - F(n+1) = Sum_{k=0..n} F(k)*J(n-k), where J=A001045, F=A000045. - Yuchun Ji, Mar 05 2019

EXAMPLE

a(2) =   0 + 2*0  +  1 =   1

a(3) =   1 + 2*0  +  1 =   2

a(4) =   2 + 2*1  +  2 =   6

a(5) =   6 + 2*2  +  3 =  13

a(6) =  13 + 2*6  +  5 =  30

a(7) =  30 + 2*13 +  8 =  64

a(8) =  64 + 2*30 + 13 = 137

a(9) = 137 + 2*64 + 21 = 286

... - Philippe Deléham, Mar 06 2013

MAPLE

with(combstruct):

TSU := [T, { T = Sequence(S, card > 1), S = Sequence(U, card > 0), U = Sequence(Z, card > 1)}, unlabeled]:

seq(count(TSU, size = j+2), j=0..32); # Peter Luschny, Jan 04 2020

MATHEMATICA

LinearRecurrence[{2, 2, -3, -2}, {0, 0, 1, 2}, 40] (* G. C. Greubel, Mar 06 2019 *)

PROG

(PARI) my(x='x+O('x^40)); concat([0, 0], Vec(x^2/((1-x-x^2)*(1-x-2*x^2)))) \\ G. C. Greubel, Mar 06 2019

(MAGMA) I:=[0, 0, 1, 2]; [n le 4 select I[n] else 2*Self(n-1) + 2*Self(n-2) -3*Self(n-3) -2*Self(n-4): n in [1..40]]; // G. C. Greubel, Mar 06 2019

(Sage) (x^2/((1-x-x^2)*(1-x-2*x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Mar 06 2019

(GAP) a:=[0, 0, 1, 2];; for n in [5..40] do a[n]:=2*a[n-1]+2*a[n-2] - 3*a[n-3]-2*a[n-4]; od; a; # G. C. Greubel, Mar 06 2019

CROSSREFS

Sequence in context: A289048 A297388 A115217 * A336875 A219753 A239305

Adjacent sequences:  A094684 A094685 A094686 * A094688 A094689 A094690

KEYWORD

easy,nonn

AUTHOR

Paul Barry, May 19 2004

STATUS

approved

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Last modified July 24 18:34 EDT 2021. Contains 346273 sequences. (Running on oeis4.)